# Distance-Uniform Graphs with Large Diameter

## TL;DR

This paper constructs large-diameter, epsilon-distance-uniform graphs, disproving a prior conjecture that their critical distance is logarithmic in size, and establishes tight bounds on this distance.

## Contribution

It provides the first construction of epsilon-distance-uniform graphs with super-logarithmic diameter, and proves this bound is optimal.

## Key findings

- Existence of epsilon-distance-uniform graphs with exponential diameter in log n
- Disproof of the conjecture that diameter is at most logarithmic in n
- Optimality of the exponential upper bound on diameter

## Abstract

An $\epsilon$-distance-uniform graph is one in which from every vertex, all but an $\epsilon$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in an $\epsilon$-distance-uniform graph with $n$ vertices. We show that for $\frac1n \le \epsilon \le \frac1{\log n}$, there exist $\epsilon$-distance-uniform graphs with critical distance $2^{\Omega(\frac{\log n}{\log \epsilon^{-1}})}$, disproving a conjecture of Alon et al. that $d$ can be at most logarithmic in $n$. We also show that our construction is best possible, in the sense that an upper bound on $d$ of the form $2^{O(\frac{\log n}{\log \epsilon^{-1}})}$ holds for all $\epsilon$ and $n$.